Skip to main content
SpringerLink
Log in

An approach for choosing discretization schemes and grid size based on the convection-diffusion equation

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A new approach for selecting proper discretization schemes and grid size is presented. This method is based on the convection-diffusion equation and can provide insight for the Navier-Stokes equation. The approach mainly addresses two aspects, i.e., the practical accuracy of diffusion term discretization and the behavior of high wavenumber disturbances. Two criteria are included in this approach. First, numerical diffusion should not affect the theoretical diffusion accuracy near the length scales of interest. This is achieved by requiring numerical diffusion to be smaller than the diffusion discretization error. Second, high wavenumber modes that are much smaller than the length scales of interest should not be amplified. These two criteria provide a range of suitable scheme combinations for convective flux and diffusive flux and an ideal interval for grid spacing. The effects of time discretization on these criteria are briefly discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pereira, J. M. C. and Pereira, J. C. F. Fourier analysis of several finite difference schemes for the one-dimensional unsteady convection-diffusion equation. International Journal for Numerical Methods in Fluids, 36(4), 417–439 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Wang, Y. T., Sun, Y., and Wang, G. X. Numerical analysis of the effect of discrete accuracy of turbulence model on numerical simulation (in Chinese). Acta Aeronautica et Astronautica Sinica, 36(5), 1453–1459 (2015)

    Google Scholar 

  3. Suzuki, H., Nagata, K., Sakai, Y., Hayase, T., Hasegawa, Y., and Ushijima, T. An attempt to improve accuracy of higher-order statistics and spectra in direct numerical simulation of incompressible wall turbulence by using the compact schemes for viscous terms. International Journal for Numerical Methods in Fluids, 73(6), 509–522 (2013)

    Article  MathSciNet  Google Scholar 

  4. Fu, D., Ma, Y. W., and Li, X. L. Direct Numerical Simulation of Compressible Turbulent Flow (in Chinese), Science Press, Beijing, 99–127 (2010)

    Google Scholar 

  5. Li, L., Yu, C. P., Zhe, C., and Li, X. L. Resolution-optimised nonlinear scheme for secondary derivatives. International Journal of Computational Fluid Dynamics, 30(2), 1–13 (2016)

    Article  MathSciNet  Google Scholar 

  6. Fu, D. X. and Ma, Y. W. A high-order-accurate difference scheme for complex flow fields. Journal of Computational Physics, 134(1), 1–15 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mattsson, K., Savard, M., and Shoeybi, M. Stable and accurate schemes for the compressible Navier-Stokes equations. Journal of Computational Physics, 227(4), 2293–2316 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mohammadian, A. Numerical approximation of viscous terms in finite volume models for shallow waters. International Journal for Numerical Methods in Fluids, 63(5), 584–599 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Kalita, J. C., Dalal, D. C., and Dass, A. K. A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients. International Journal for Numerical Methods in Fluids, 38(12), 1111–1131 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sen, S. A new family of (5,5)CC-4OC schemes applicable for unsteady Navier-Stokes equations. Journal of Computational Physics, 251(23), 251–271 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, H. X., Guo, C., and Zong, W. G. Problems about grid and high order schemes (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 31(4), 398–405 (1999)

    Google Scholar 

  12. Fu, D. X. and Ma, Y. W. High order accurate schemes and numerical simulation of multi-scale structures in complex flow fields (in Chinese). Acta Aerodynamica Sinica, 16(1), 24–35 (1998)

    Google Scholar 

  13. Wen, J. and Gao, Z. H. Numerical analysis of high-order difference schemes of Burgers equation (in Chinese). Aeronautical Computing Technique, 38(3), 74–77 (2008)

    Google Scholar 

  14. Tu, G. H., Deng, X. G., and Mao, M. L. A staggered non-oscillatory finite difference method for high-order discretization of viscous terms (in Chinese). Acta Aerodynamica Sinica, 29(1), 10–15 (2011)

    Google Scholar 

  15. Lele, S. K. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103(1), 16–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, Q., Fu, D. X., and Ma, Y. W. Behavior analysis of upwind compact difference schemes for the fully-discretized convection equation (in Chinese). Chinese Journal of Computational Physics, 16(5), 489–495 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Lin Zhou, Zhenghong Gao & Yuan Gao

Authors
  1. Lin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenghong Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenghong Gao.

Additional information

Citation: Zhou, L., Gao, Z. H., and Gao, Y. An approach for choosing discretization schemes and grid size based on the convection-diffusion equation. Applied Mathematics and Mechanics (English Edition) (2018) https://doi.org/10.1007/s10483-018-2341-9

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, L., Gao, Z. & Gao, Y. An approach for choosing discretization schemes and grid size based on the convection-diffusion equation. Appl. Math. Mech.-Engl. Ed. 39, 877–890 (2018). https://doi.org/10.1007/s10483-018-2341-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-018-2341-9

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation